On the construction of certain 6-dimensional symplectic manifolds with Hamiltonian circle actions

Let $(M,\omega )$ be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point set consists of isolated points or surfaces. Assume dim $H^2(M)<3$, in \cite{L}, we defined a certain invariant of such spaces which consists of fixed point data and twist type, and we divided the possible values of these invariants into six types''. In this paper, we construct such manifolds with these types’'. As a consequence, we have a precise list of the values of these invariants.